Does this hyperparameter optimisation approach yield the optimal hyperparameters?
After you've computed $h^{1}_{optimal}$ the only thing you can be sure is that this is the best (assuming constrained case) value of $h^1$ (with respect to some model performance metric) given your initial values for $h^2, ..., h^n$. If you change a bit any of $h^2, ..., h^n$ you're no longer certain that the value $h^1$ you found is the optimal one. So yes, the key here is the assumption about the independence.